Optimal. Leaf size=39 \[ -\frac {x}{3}+\frac {x^2}{6}-\frac {x^3}{9}+\frac {1}{3} \log (1+x)+\frac {1}{3} x^3 \log (1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2442, 45}
\begin {gather*} -\frac {x^3}{9}+\frac {1}{3} x^3 \log (x+1)+\frac {x^2}{6}-\frac {x}{3}+\frac {1}{3} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rubi steps
\begin {align*} \int x^2 \log (1+x) \, dx &=\frac {1}{3} x^3 \log (1+x)-\frac {1}{3} \int \frac {x^3}{1+x} \, dx\\ &=\frac {1}{3} x^3 \log (1+x)-\frac {1}{3} \int \left (1+\frac {1}{-1-x}-x+x^2\right ) \, dx\\ &=-\frac {x}{3}+\frac {x^2}{6}-\frac {x^3}{9}+\frac {1}{3} \log (1+x)+\frac {1}{3} x^3 \log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.72 \begin {gather*} \frac {1}{18} \left (x \left (-6+3 x-2 x^2\right )+6 \left (1+x^3\right ) \log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 50, normalized size = 1.28
method | result | size |
meijerg | \(-\frac {x \left (4 x^{2}-6 x +12\right )}{36}+\frac {\left (4 x^{3}+4\right ) \ln \left (1+x \right )}{12}\) | \(28\) |
norman | \(-\frac {x}{3}+\frac {x^{2}}{6}-\frac {x^{3}}{9}+\frac {\ln \left (1+x \right )}{3}+\frac {x^{3} \ln \left (1+x \right )}{3}\) | \(30\) |
risch | \(-\frac {x}{3}+\frac {x^{2}}{6}-\frac {x^{3}}{9}+\frac {\ln \left (1+x \right )}{3}+\frac {x^{3} \ln \left (1+x \right )}{3}\) | \(30\) |
derivativedivides | \(\frac {\left (1+x \right )^{3} \ln \left (1+x \right )}{3}-\frac {\left (1+x \right )^{3}}{9}-\ln \left (1+x \right ) \left (1+x \right )^{2}+\frac {\left (1+x \right )^{2}}{2}+\left (1+x \right ) \ln \left (1+x \right )-1-x\) | \(50\) |
default | \(\frac {\left (1+x \right )^{3} \ln \left (1+x \right )}{3}-\frac {\left (1+x \right )^{3}}{9}-\ln \left (1+x \right ) \left (1+x \right )^{2}+\frac {\left (1+x \right )^{2}}{2}+\left (1+x \right ) \ln \left (1+x \right )-1-x\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.78, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (x + 1\right ) - \frac {1}{9} \, x^{3} + \frac {1}{6} \, x^{2} - \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 25, normalized size = 0.64 \begin {gather*} -\frac {1}{9} \, x^{3} + \frac {1}{6} \, x^{2} + \frac {1}{3} \, {\left (x^{3} + 1\right )} \log \left (x + 1\right ) - \frac {1}{3} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 29, normalized size = 0.74 \begin {gather*} \frac {x^{3} \log {\left (x + 1 \right )}}{3} - \frac {x^{3}}{9} + \frac {x^{2}}{6} - \frac {x}{3} + \frac {\log {\left (x + 1 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.52, size = 49, normalized size = 1.26 \begin {gather*} \frac {1}{3} \, {\left (x + 1\right )}^{3} \log \left (x + 1\right ) - \frac {1}{9} \, {\left (x + 1\right )}^{3} - {\left (x + 1\right )}^{2} \log \left (x + 1\right ) + \frac {1}{2} \, {\left (x + 1\right )}^{2} + {\left (x + 1\right )} \log \left (x + 1\right ) - x - 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 25, normalized size = 0.64 \begin {gather*} \frac {x^2}{6}-\frac {x}{3}-\frac {x^3}{9}+\frac {\ln \left (x+1\right )\,\left (x^3+1\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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